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d.Precalculus with Limits, Answers to Section 3.1 1
Chapter 3Section 3.1 (page 226)
Vocabulary Check (page 226)1. algebraic 2. transcendental
3. natural exponential; natural 4.
5.
1. 946.852 2. 3.488 3. 0.006 4. 0.5445. 1767.767 6.7. d 8. c 9. a 10. b
11.
12.
13.
14.
15.
16.
17. Shift the graph of f four units to the right.18. Shift the graph of f one unit upward.19. Shift the graph of f five units upward.20. Reflect the graph of f in the y-axis and shift three units to
the right.21. Reflect the graph of f in the x-axis and y-axis and shift six
units to the right.22. Reflect f in the x-axis and shift five units upward.23. 24.
−3 3
−1
3
−3
−1
3
3
x
7
6
5
4
2
1
1−1−2−3 2 3 4 5
y
x321−1−2−3
5
4
3
2
1
−1
y
x
5
4
3
321−1−2−3
2
1
−1
y
x321−1−2−3
5
4
3
1
−1
y
x321−1−2−3
5
4
3
2
−1
y
x321−1−2−3
5
4
3
2
1
−1
y
1.274 � 1025
A � Pert
A � P�1 �rn�
nt
x 0 1 2
4 2 1 0.5 0.25f �x�
�1�2
0 1 2
36 6 1 0.167 0.028f �x�
�1�2x
x 0 1 2
0.25 0.5 1 2 4f �x�
�1�2
0 1 2
0.028 0.167 1 6 36f �x�
�1�2x
x 0 1 2
0.125 0.25 0.5 1 2f �x�
�1�2
x 0 1 2 3
3.004 3.016 3.063 3.25 4f �x�
�1
333202CB03_AN.qxd 4/13/06 5:38 PM Page 1
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d.
Precalculus with Limits, Answers to Section 3.1 2
(Continued)
25. 26.
27. 0.472 28. 24.533 29.30. 31. 7166.647 32. 679.57033.
34.
35.
36.
37.
38.
39. 40.
41. 42.
43. 44.
−2 40
4
−30
3
4
−16 17
−2
20
−100
23
22
−4 8
−2
6
−7
−1
5
7
x
8
7
6
5
4
3
1
87654321−1
y
x7654321
9
8
7
6
5
3
2
1
−1−2−3
y
x
6
5
4
3
2
1
−14321−1−2−3
y
x−1−2−3−4−5−6−7−8 1
8
7
6
5
4
3
2
1
y
x
5
4
3
2
1
−1321−1−2−3
y
x321−1−2−3
5
4
3
2
1
−1
y
1.956 � 1052
3.857 � 10�22
−6 3
−3
3
−10
5
4
x 0 1 2
0.135 0.368 1 2.718 7.389f �x�
�1�2
x
0.055 0.149 0.406 1.104 3f �x�
�4�5�6�7�8
0 1 2
7.389 2.718 1 0.368 0.135f �x�
�1�2x
0 1 2
5.437 3.297 2 1.213 0.736f �x�
�1�2x
x 0 1 2
4.037 4.100 4.271 4.736 6f �x�
�1�2
0 2 4 5 6
2.007 2.050 2.368 3 4.718f �x�
x
333202CB03_AN.qxd 4/13/06 5:38 PM Page 2
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d.Precalculus with Limits, Answers to Section 3.1 3
(Continued)
45. 46. 47.48. 49. 50.51. 52.53.
54.
55.
56.
57.
58.
59.
60.
61. $222,822.57 62. $212,605.41 63. $35.4564. (a) (b) $421.12
(c) $350
65. (a) (b)(c)
66. (a) Decreasing(b)
(c)
67. (a) 25 grams (b) 16.21 grams(c)
68. (a) 10 grams (b) 7.85 grams(c)
69. (a)
(b)
(c) 63.14% (d) 38 masses
00 120
110
Time (in years)
t80004000
2
4
6
8
10
12
Q
Mas
s of
14C
(in
gra
ms)
00
5000
30
P�20� � 140.84 millionP�10� � 146.44 millionP�8� � 147.58 million
V�2� � 1,000,059.6V�1.5� � 100,004.47V�1� � 10,000.298
00
2000
1200
x � 2, 3x � 3, �1x �
52x �
13x � �4
x � �3x � 7x � 2
n 1 2 4
A $3200.21 $3205.09 $3207.57
n 12 365 Continuous
A $3209.23 $3210.06 $3210.06
n 1 2 4
A $4515.28 $4535.05 $4545.11
n 12 365 Continuous
A $4551.89 $4555.18 $4555.30
t 10 20 30
A $17,901.90 $26,706.49 $39,841.40
t 40 50
A $59,436.39 $88,668.67
t 10 20 30
A $22,986.49 $44,031.56 $84,344.25
t 40 50
A $161,564.86 $309,484.08
n 1 2 4 12
A $1480.24 $1485.95 $1488.86 $1490.83
n 365 Continuous
A $1491.79 $1491.82
n 1 2 4
A $10,285.72 $10,640.89 $10,828.46
n 12 365 Continuous
A $10,957.45 $11,021.00 $11,023.18
t 10 20 30
A $21,865.43 $39,841.40 $72,595.77
t 40 50
A $132,278.12 $241,026.44
t 10 20 30
A $17,028.81 $24,165.03 $34,291.81
t 40 50
A $48,662.40 $69,055.23
x 0 25 50 75 100
Model 12.5 44.5 81.82 96.19 99.3
Actual 12 44 81 96 99
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d.
Precalculus with Limits, Answers to Section 3.1 4
(Continued)
70. (a) (b) 32,357 pascals
71. True. As but never reaches 72. False. e is an irrational number.73. 74.75. 76. None are equal.77. (a) (b)
78. (a)
Decreasing: Increasing: Relative maximum: Relative minimum:
(b)
Decreasing: Increasing: Relative maximum:
79.
As As
80. c, d
81.
82. and
83. 84.
85. Answers will vary.
y
x64 8
−6
−4
−2
2
4
6
2−2−4x3−3−15−18 −6
9
12
6
3
−9
−3
−6
y
x ≥ 2y � ��x � 2�,y � x � 2
y � ±�25 � x2
f �x� → g�x�.x → ��,f �x� → g�x�.x →�,
−3 30
g
f
4
�1.44, 4.25����, 1.44��1.44, ��
−2
−2
10
6
�0, 0��2, 4e�2�
�0, 2��2, �����, 0�,
−2 7
−1
5
x > 0x < 0
x21−1−2
−1
1
2
3
y
y = 4x y = 3x
f �x� � g�x� � h�x�g�x� � h�x�f �x� � h�x�
�2.f �x� → �2x → ��,
Altitude (in km)
Atm
osph
eric
pre
ssur
e(i
n pa
scal
s)P
h5 10 15 20
40,000
60,000
80,000
25
100,000
120,000
20,000
333202CB03_AN.qxd 4/13/06 5:38 PM Page 4
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d.Precalculus with Limits, Answers to Section 3.2 5
Section 3.2 (page 236)
Vocabulary Check (page 236)1. logarithmic 2. 10 3. natural; e4. 5.
1. 2. 3.4. 5. 6.7. 8. 9.
10. 11. 12.
13. 14. 15.
16. 17. 4 18. 19. 0
20. 1 21. 2 22. 23.24. 25. 1.097 26. 1.877 27. 4
28. 0 29. 1 30. 15
31. Domain: x-intercept: Vertical asymptote:
32. Domain: x-intercept: Vertical asymptote:
33. Domain: x-intercept: Vertical asymptote:
34. Domain:
x-intercept: Vertical asymptote:
35. Domain: x-intercept: Vertical asymptote:
36. Domain:
x-intercept:
Vertical asymptote:
37. Domain: x-intercept: Vertical asymptote:
38. Domain:
x-intercept: Vertical asymptote:
39. c 40. f 41. d
42. e 43. b 44. a
45. 46.
47. 48.
49. 50.
51. 52.
53. 54.
55. 56.
57.
58.
59. 60. 61. 2.913
62. 63. 64. 0.693
65. 3 66. 67. 68. �52�
23�2
�0.575�3.418
ln 3 � 2xln 4 � x
ln 0.0165 . . . � �4.1
ln 0.6065 . . . � �0.5
ln 1.3956 . . . �13ln 1.6487 . . . �
12
ln 7.3890 . . . � 2ln 20.0855 . . . � 3
e1 � ee0 � 1
e6.520 . . . � 679e5.521 . . . � 250
e2.302 . . . � 10e1.386. . . � 4
e�0.916 . . . �25e�0.693 . . . �
12
x � 0��1, 0�
���, 0�
−3 −2 −1 1
−2
−1
1
2
x
y
x � 0�5, 0�
�0, ��
4 6 8
−4
−2
2
4
x
y
x � 1
�626625, 0�
�1, ��
2 3 4 5 6
1
2
3
4
5
6
x
y
x � �2��1, 0�
��2, ��
6
−4
−2
2
4
x
y
x � 3�4, 0�
�3, ��
2 4 6 8 10
−4
−2
2
4
6
x
y
x � 0�9, 0�
�0, ��
2 4 6 8 10 12
−6
−4
−2
2
4
6
x
y
x � 0�1, 0�
�0, ��
1−1 2 3
−2
−1
1
2
x
y
x � 0�1, 0�
�0, ��
−1 1 2 3
−2
−1
1
2
x
y
�2.699�0.097�3
12log 0.001 � �3
log7 1 � 0log4 164 � �3log6
136 � �2
log9 27 �32log81 3 �
14log8 64 � 2
log5 125 � 382�3 � 4361�2 � 6163�4 � 8322�5 � 410�3 �
11000
7�2 �1
4934 � 8143 � 64
x � yaloga x � x
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d.
Precalculus with Limits, Answers to Section 3.2 6
(Continued)
69. Domain: x-intercept: Vertical asymptote:
70. Domain:
x-intercept: Vertical asymptote:
71. Domain: x-intercept: Vertical asymptote:
72. Domain:
x-intercept: Vertical asymptote:
73. 74.
75. 76.
77. 78.
79. 80. 81.
82. 83. 84.
85. 86.
87. (a) 30 years; 20 years (b) $396,234; $301,123.20(c) $246,234; $151,123.20(d) The monthly payment must be greater than
$1000.
88. (a)
The number of years required to multiply the originalinvestment by K increases with K. However, the largerthe value of K, the fewer the years required to increasethe value of the investment by an additional multipleof the original investment.
(b)
89. (a)
(b) 80 (c) 68.1 (d) 62.3
90. (a) 120 decibels(b) 100 decibels(c) No, the difference results from the logarithmic rela-
tionship between intensity and number of decibels.
91. False. Reflecting about the line will determinethe graph of
92. True.
93. 94.
The functions f and g The functions f and gare inverses. are inverses.
−2 −1 1 2
−2
−1
1
2
x
g
f
y
−2 −1 1 2
−2
−1
1
2
x
g
f
y
log3 27 � 3 ⇒ 33 � 27
f �x�.y � xg�x�
00 12
100
2 4 6 8 10 12
5
10
15
20
25
K
t
x � 1000;
x � �2, 3x � �5, 5
x � 6x � 4x �95
x � 7x � 12x � 3
−5 10
−6
4
0
−1
9
5
−4 5
−3
3
0
−3
9
3
−1 5
−2
2
−1
−2
5
2
x � 3�2, 0�
���, 3�
−2 −1 1 2 4
−3
−2
−1
2
3
x
y
x � 0��1, 0�
���, 0�
−3 −2 −1 1
−2
1
2
x
y
x � �1�0, 0�
��1, ��
2−2 4 6 8x
6
4
2
y
x � 1�2, 0�
�1, ��y
x32 4 5
−3
−2
−1
1
2
3
1−1
K 1 2 4 6 8 10 12
t 0 7.3 14.6 18.9 21.9 24.2 26.2
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d.Precalculus with Limits, Answers to Section 3.2 7
(Continued)
95. 96.
The functions f and g The functions f and gare inverses. are inverses.
97. (a) (b)
The natural log The natural log function grows at a function grows at a slower rate than the slower rate than thesquare root function. fourth root function.
98. (a)
(b) 0(c)
99. (a) False (b) True (c) True (d) False
100. Answers will vary.
101. (a) (b) Increasing: Decreasing:
(c) Relative minimum:
102. (a) (b) Increasing: Decreasing:
(c) Relative minimum:
103. 15 104. 1 105. 4300
106. 107. 1028 108. �344�2
�0, 0�
���, 0��0, ��
−9
−4
9
8
�1, 0�
�0, 1��1, ��
−1
−2
8
4
00
100
0.5
g�x�;g�x�;
00
20,000
g
f
15
00
1000
f
g
40
−2 −1 1 2
−2
−1
1
2
x
g
f
y
−2 −1 1 2
−2
−1
1
2
x
g
f
y
x 1 5 10
0 0.322 0.230 0.046f �x�
102
x
0.00092 0.0000138f �x�
106104
The natural log func-tion grows at a slower rate than the fourth root function.
g�x�;The natural log function grows at a slowerrate than the square rootfunction.
g�x�;
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d.
Precalculus with Limits, Answers to Section 3.3 8
Section 3.3 (page 243)
Vocabulary Check (page 243)
1. change-of-base 2.
3. c 4. a 5. b
1. (a) (b) 2. (a) (b)
3. (a) (b) 4. (a) (b)
5. (a) (b) 6. (a) (b)
7. (a) (b) 8. (a) (b)
9. 1.771 10. 0.712 11. 12.
13. 14. 15. 2.633 16.
17. 18. 19.
20. 21. 22. 23. 2
24. 25. 26. 27. 2.4 28.
29. is not in the domain of
30. is not in the domain of
31. 4.5 32. 12 33. 34. 35. 7
36. 7 37. 2 38. 3 39.
40. 41. 42.
43. 44. 45.
46. 47.
48. 49.
50.
51. 52.
53. 54.
55.
56.
57.
58.
59. 60.
61. 62. 63. 64.
65. 66.
67. 68. 69.
70. 71. 72.
73. 74.
75. 76.
77. 78.
79. Property 2
80.
Properties 1 and 3
81. 60 dB 82. 24 dB difference
83.
84. (a) (b) 90(c) 79.5 (d) 73.3(e)
(f) 9 months (g)
85.
86. (a) and (b)
(c)
The results are similar.(d)
(e) Answers will vary.
87. False. 88. True; Property 1
89. False.
90. False.
91. False. 92. True
93–94. Answers will vary.
u � v 2
f ��x � �12 f �x�
ln�x � 2� � ln x � ln 2
ln 1 � 0
T � 21 �1
0.001t � 0.016
00
30
0.07
T � 21 � e�0.037t�3.997
00
30
5
00
30
80
y � 256.24 � 20.8 ln x
90 � 15 log 10 � 75
120
95
70
90 � log�t � 1�15
� 3
� 10�log I � 12�;
� 12 � log7�10;
log7�70 �12 �log7 7 � log7 10�
log2 324 � log2 32 � log2 4;
log4 x6�x � 1��x � 1log8
3�y�y � 4�2
y � 1
ln� x3
x2 � 1�2
ln 3�x�x � 3�2
x2 � 1
ln z4�z � 5�4
�z � 5�2lnx
�x2 � 4�4
log3 x3y4
z4log xz3
y2ln 64�z � 4�5
lnx
�x � 1�3log6
116x 4log3 4�5x
log7�z � 2�2�3log2�x � 4� 2
log5 8t
log4
z
yln ytln 3x
ln x �12 ln�x � 2�3
4 ln x �14 ln�x2 � 3�
log x � 4 log y � 5 log z
2 log5 x � 2 log5 y � 3 log5 z
12 log2 x � 4 log2 y � 4 log2 z
4 ln x �12 ln y � 5 ln z
ln x �32 ln y1
3 ln x �13 ln y
ln 6 �12 ln�x 2 � 1�1
2 log2�a � 1� � 2 log2 3
ln�x � 1� � ln�x � 1� � 3ln x
ln z � 2 ln�z � 1�log 4 � 2 log x � log y
ln x � ln y � 2 ln z13 ln t
12 ln z�3 log6 z1 � log5 x
log y � log 24 log8 xlog3 10 � log3 z
log4 5 � log4 x
34�
12
log2 x.�16
log3 x.�9
�0.813
34�3
ln 6 � 26 � ln 5log 3 � 2
�3 � log5 24 � 4 log2 332
�3.823�0.694�0.417
�1.161�2.000
ln xln 7.1
log xlog 7.1
ln xln 2.6
log xlog 2.6
ln 34ln x
log 34log x
ln 3
10
ln x
log310
log x
ln xln 13
log xlog 13
ln xln 1
5
log xlog
15
ln xln 3
log xlog 3
ln xln 5
log xlog 5
log xlog a
�ln xln a
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d.Precalculus with Limits, Answers to Section 3.3 9
(Continued)
95. 96.
97. 98.
99.
100.
101.
102.
103. 104. 105.
106. 107. 108.
109. 110.1 ± �31
2�1 ± �97
6
1, 14
�1, 13
�xy�2
x � y
1, x � 0, y � 027y 3
8x 6
3x4
2y 3, x � 0
ln 9 � 2.1972
ln 20 � 2.9956ln 8 � 2.0793
ln 18 � 2.8903ln 6 � 1.7917
ln 16 � 2.7724ln 5 � 1.6094
ln 15 � 2.7080ln 4 � 1.3862
ln 12 � 2.4848ln 3 � 1.0986
ln 10 � 2.3025ln 2 � 0.6931
1 2 3 4
−2
−1
1
2
x
f = hg
y
f �x� � h�x�; Property 2
−1 5
−2
2
f �x� �log x
log 12.4�
ln xln 12.4
−1
−2
5
2
f �x� �log x
log11.8�
ln xln 11.8
−1 5
−2
2
−3
−3
6
3
f �x� �log xlog 14
�ln xln 14
f �x� �log xlog 1
2
�ln xln 12
−1 5
−2
2
−3
−3
6
3
f �x� �log xlog 4
�ln xln 4
f �x� �log xlog 2
�ln xln 2
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d.
Precalculus with Limits, Answers to Section 3.4 10
Section 3.4 (page 253)
Vocabulary Check (page 253)1. solve2. (a) (b) (c) x (d) x3. extraneous
1. (a) Yes (b) No 2. (a) No (b) No3. (a) No (b) Yes (c) Yes, approximate4. (a) Yes (b) No (c) Yes, approximate5. (a) Yes, approximate (b) No (c) Yes6. (a) Yes (b) No (c) No7. (a) No (b) Yes (c) Yes, approximate8. (a) Yes (b) Yes, approximate (c) No9. 2 10. 5 11. 12. 13. 2 14. 5
15. 16. 17.18. 19. 64 20. 0.00821. 22. 23.24. 25. 26.
27. 28. 0, 1
29. 30.
31. 32.
33. 34.
35. 36. 37. 2
38. 39. 4 40. 8
41. 42.
43. 44.
45. 46.
47. 48.
49. 50. 51. 0
52. 53.
54. 55.
56.
57. 58. No solution
59. 60.
61. 62.
63. 64.
65. 66.
67. 68.
69. 70.
3.847
71. 72.
12.207 1.081
73. 74.
16.636 1.236
75. 76. 77.
78. 79. 1,000,000 80.
81. 82.
83. 84.85. 86.87. 88.
89. No solution 90.
91. 92.
93. No solution 94. 95. 7
96. No solution 97. 98. 2
99. 2 100. 9 101.
102.1225 � 125�73
2� 1146.500
725 � 125�338
� 180.384
�1 � �172
� 1.562
3 � �132
� 3.303
�3��9 � 4e2
� 0.7291 � �1 � e � 2.928
�1 ��1� 4e2
� 1.223
1011�5 � 2 � 160.4892�311�6� � 14.988e�4�3 � 0.264e�2�3 � 0.513e10 � 8 � 22,034.466e2 � 2 � 5.389
e7�2 � 33.115e10�3
5� 5.606
1003
� 33.333e4
� 0.680
e2.4
2� 5.512e2 � 7.389e�3 � 0.050
−5 5
−35
10
−40 40
−10
2
−5 5
−7
13
−20 40
−4
8
�0.478
−3 7
−15
5
−6 9
−1200
300
�2.322�0.427
−5 5
−20
20
−6 15
−30
6
ln 30
3 ln�16 � 0.87826 � � 0.409
ln 2
12 ln�1 �0.1012 � � 6.960
ln 219 ln 3.938225
� 0.247ln 4
365 ln�1 �0.065365 � � 21.330
ln 316
� 0.57212
ln 1498 � 3.656
ln 7 � 1.9462 ln 75 � 8.635
ln 4 � 1.386
ln 2 � 0.693; ln 3 � 1.099
ln 5 � 1.6093 �ln 72
2 ln 4� 2.548
ln 833 ln 2
�13
� 0.805ln 253
� 2.120
�14
ln 3
40� 0.648�ln
35
� 0.511
ln 502
� 1.956ln 12
3� 0.828
6 �ln 5ln 3
� 4.5351 �ln 7ln 5
� 2.209
6 � log 75
� 6.14613 log�3
2� � 0.059
�ln 64 � ln 431ln 8
� �4.9173 �ln 565
ln 2� �6.142
�ln�0.10�3 ln 4
� 0.554
ln 3000
5 ln 6� 0.894
ln 80
2 ln 3� 1.994
ln 37ln 6
� 2.015ln 28 � 3.332
ln 914
� 3.125ln 5 � 1.609
ln 16ln 5
� 1.723ln 5
ln 3� 1.465
� �0.618� 1.618,
�2, 42, �1�5, 0��9, 2��2
3, 9��3, 8�e�7 � 0.000912
e �1 � 0.368ln 4 � 1.386ln 2 � 0.693�3�5
x � yx � y
333202CB03_AN.qxd 4/13/06 5:38 PM Page 10
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d.Precalculus with Limits, Answers to Section 3.4 11
(Continued)
103. 104.
2.807 2.197105. 106.
20.086 14.182107. (a) 8.2 years (b) 12.9 years108. (a) 5.8 years (b) 9.2 years109. (a) 1426 units (b) 1498 units110. (a) 303 units (b) 528 units111. (a)
(b) The yield will approach 6.7 million cubicfeet per acre.
(c) 29.3 years112. 12.76 inches 113. 2001 114. 2001115. (a) and The range falls between 0% and
100%.(b) Males: 69.71 inches Females: 64.51 inches
116. (a)
(b) Horizontal asymptotes: The proportion of correct responses will approach0.83 as the number of trials increases.
(c) trials117. (a)
(b)
The model appears to fit the data well.
(c) 1.2 meters(d) No. According to the model, when the number of ’s
is less than 23, is between 2.276 meters and 4.404meters, which isn’t realistic in most vehicles.
118. (a) ; Room temperature (b) hour
119.True by Property 1 in Section 5.3.
120.False.
121.False.
122.
True by Property 2 in Section 5.3.
123. Yes. See Exercise 93.
124. For years, double the amount you invest. Foryears, double your interest rate or double the
number of years, because either of these will double theexponent in the exponential function.
125. Yes. Time to double:
Time to quadruple:
126. Answers will vary. 127.
128. 129. 130.
131. 132.
133. 134.
135. 1.226 136. 1.262 137. 138. 1.486�5.595
y
x64
−6
−2
1
4
6
2−2−4−6
y
x−1−2−3−4 1 3 4
−3
1
2
3
4
5
y
x−2−4−6 2 4 6 8
−2
−4
−6
2
4
6
8
y
x−2−4−6−8 2 4 6 8
−2
2
4
6
8
12
14
12�10 � 15 3�34�2 � 10
4�x�y2�3y
t �ln 4
r� 2�ln 2
r �t �
ln 2r
;
rt > ln 2rt < ln 2
logb uv
� logb u � logb v
1.95 � log�100 � 10� � log 100 � log 10 � 1
logb�u � v� � logb u � logb v
2.04 � log�10 � 100� � �log 10��log 100� � 2
logb�u � v� � �logb u��logb v�
logb uv � logb u � logb v
� 0.81T � 20
xg
00
1.2
200
� 5
P � 0.83P � 0,
00 40
1.0
y � 0;y � 100
V � 6.7;
00
1500
10
−5
−3
30
18
−5
−1
30
5
−2
−200
10
800
−8 10
−2
10
x 0.2 0.4 0.6 0.8 1.0
y 162.6 78.5 52.5 40.5 33.9
333202CB03_AN.qxd 4/13/06 5:38 PM Page 11
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d.
Precalculus with Limits, Answers to Section 3.5 12
Section 3.5 (page 264)
Vocabulary Check (page 264)1.2.3. normally distributed 4. bell; average value5. sigmoidal
1. c 2. e 3. b 4. a 5. d 6. fInitial Annual Time to Amount AfterInvestment % Rate Double 10 years
7. $1000 3.5% 19.8 yr $1419.07
8. $750 10.5% 6.60 yr $2143.24
9. $750 8.9438% 7.75 yr $1834.33
10. $10,000 5.7762% 12 yr $17,817.93
11. $500 11.0% 6.3 yr $1505.00
12. $600 34.66% 2 yr $19,205.00
13. $6376.28 4.5% 15.4 yr $10,000.00
14. $1637.46 2% 34.7 yr $2000.00
15. $112,087.09 16. $4214.16
17. (a) 6.642 years (b) 6.330 years(c) 6.302 years (d) 6.301 years
18. (a) 6.94 years (b) 6.63 years(c) 6.602 years (d) 6.601 years
19.
20.
Use PwrReg:
21.
22.
Use PwrReg:
23. 24.
Continuous compoundingDaily compounding
Half-life Initial Amount After(years) Quantity 1000 Years
25. 1599 10 g 6.48 g26. 1599 2.31 g 1.5 g27. 5715 2.26 g 2 g28. 5715 3 g 2.66 g29. 24,100 2.16 g 2.1 g30. 24,100 0.41 g 0.4 g31. 32.33. 34.35. (a) Decreasing due to the negative exponent.
(b) 2000: population of 2430 thousand2003: population of 2408.95 thousand
(c) 201836. (a) Bulgaria:
Canada: China: United Kingdom: United States:
(b) b; The greater the rate of growth, the greater the value of b.
(c) b determines whether the population is increasingor decreasing
37.38. 39. 3.15 hours40. 61.16 hours41. (a) (b)42. 15,642 years43. (a) (b)
(c)
The exponential model depreciates faster.(d)
(e) Answers will vary.
0 40
32,000
V � 30,788e�0.268tV � �6394t � 30,788
� 4797 years old� 12,180 years old
k � 0.1337; $144.98 million� 5,309,734 hitsk � 0.2988;
�b < 0�.�b > 0�
y � 282.3e0.00910t; 370.9 milliony � 59.5e0.00282t; 64.8 million
y � 1268.9e0.00602t; 1520.1 milliony � 31.3e0.00915t; 41.2 milliony � 7.8e�0.00940t; 5.9 million
y � e�0.4621xy � 5e�0.4024x
y �12e0.5756xy � e 0.7675x
0
2
A t= 1 + 0.06[[ [[
A = 1 + 0.055365( ) 365t[[ [[
0 10
Am
ount
(in
dol
lars
)
t
A t= 1 + 0.075 [[ [[
A e= 0.07t
A
1.00
1.25
1.50
1.75
2.00
2 4 6 8 10
t � 1.222r�1
00
0.16
60
t � 1.099r�1
00
0.16
60
y � a � b log xy � a � b ln x;y � ae�bxy � aebx;
r 2% 4% 6% 8% 10% 12%
t 54.93 27.47 18.31 13.73 10.99 9.16
r 2% 4% 6% 8% 10% 12%
55.48 28.01 18.85 14.27 11.53 9.69t
t 1 3
24,394 11,606
23,550 13,779V � 30,788e�0.268t
V � �6394t � 30,788
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d.Precalculus with Limits, Answers to Section 3.5 13
(Continued)
44. (a) (b)(c)
The exponential model depreciates faster.(d)
(e) Answers will vary.45. (a)
(b) (c) 55,625
46. (a) (b) 36 days47. (a) (b) 100
48. (a) (b) 5.4 hours per week
49. (a) 203 animals (b) 13 years(c)
Horizontal asymptotes: The popula-tion size will approach 1000 as time increases.
50. (a) (b) 287,567 units sold
51. (a) (b)(c)
52. (a) 7.91 (b) 7.68 (c) 5.4053. (a) 20 decibels (b) 70 decibels
(c) 40 decibels (d) 120 decibels
54. (a) 10 decibels (b) 140 decibels(c) 80 decibels (d) 100 decibels
55. 95% 56. 97% 57. 4.64 58. 4.95
59. moles per liter
60. moles per liter 61.
62. 10 63. 3:00 A.M.
64. (a) (b) Interest;
(c)
Interest; years; The interest is still the majorityof the monthly payment in the early years, but now theprincipal and interest are nearly equal when years.
65. (a)
(b) years; Yes
66. (a)or
(b)
(c)
(d) Model Model Model Model The quadratic (model ) fits best.
67. False. The domain can be the set of real numbers for alogistic growth function.
t2
t4: Sum � 2.7t3: Sum � 5.6t2:: Sum � 1.1t1: Sum � 2.0
200
100
t3
t2
t1t4
25
t4 � 1.5385�1.0296�st4 � 1.5385e0.02913s
t3 � 0.2729s � 6.0143
� 210
240
150,000
t � 11
t � 11
u
v
00
20
800
t � 26 years
00
35
v
u
800
105.110�3.2 � 6.3 � 10�4
1.58 � 10�6
104.2 � 15,849108.3 � 199,526,231107.9 � 79,432,823
S �500,000
1 � 0.6e0.026t
y � 0, y � 1000.
00
40
1200
4 70
0.9
70 1150
0.04
N � 30�1 � e�0.050t�Time (in years)
Sale
s(i
n th
ousa
nds
of u
nits
)
S
t5 3025201510
30
60
90
120
S�t� � 100�1 � e�0.1625t�
0 40
1200
V � 1150e�0.368799tV � �300t � 1150
t 1 year 3 years
850 250
795 380V � 1150e�0.368799t
V � �300t � 1150
30 40 50 60 70 80 90
3.6 4.6 6.7 9.4 12.5 15.9 19.6
3.3 4.9 7.0 9.5 12.5 15.9 19.9
2.2 4.9 7.6 10.4 13.1 15.8 18.5
3.7 4.9 6.6 8.8 11.8 15.8 21.2t4
t3
t2
t1
s
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d.
Precalculus with Limits, Answers to Section 3.5 14
(Continued)
68. False. A logistic growth function never has an x-intercept.
69. False. The graph of is the graph of shifted upwardfive units.
70. True. The graph of a Gaussian model will never have an x-intercept.
71. (a) Logarithmic (b) Logistic (c) Exponential(d) Linear (e) None of the above (f) Exponential
72. Answers will vary.
73. (a) (b)(c)(d) 3
74. (a) (b)(c)(d)
75. (a) (b)
(c)
(d)
76. (a) (b) 5
(c)
(d)
77. (a) (b)
(c)
(d) 1
78. (a) (b)
(c)
(d)
79. 80.
81. 82.
83. 84.
x
−2
−4
−6
−10
64−4−6
2
−8
y
x4321−1−2−3−4
7
6
5
4
3
2
1
y
x
−5
−30
−35
2 4−4 8
y
x642
2
−2−2−4−6
y
x
3
−2
−3
1−1−2−3
2
−132
y
x8
8
6
4
2
62−2−2
10 12
10
y
16
�56, � 1
12��9.25y
x
23
13( (, −−
−1 1 2 3
−2
1
2
73
16( (,
�58, �1
8��1
8y
x1
1
12
14( (, −
34( (, 0
12
−
12
12
−
43
�172 , 2�
y
x64 8 10
−6
−4
−2
2
4
6
2−2
(7, 0)
(10, 4)
�5
11
�172 , 12��146y
x−2 2 4 6 8 10 14
−2
−4
−6
−8
2
4
6
8
(3, 3)
(14, −2)
�25
��1, �1�2�29y
x
(−6, 1)
(4, −3)
−4−6 2 4 6−2
−4
−6
2
4
6
��12, 72�
�10y
x−1−2−3 1 2 3
−1
1
2
3
5 (0, 5)
(−1, 2)
g�x�f �x�
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d.Precalculus with Limits, Answers to Section 3.5 15
(Continued)
85. 86.
87. 88.
89. 90.
91. 92.
93. Answers will vary.
x
2
1
5
1 2 3 4 5−1−2
−2
−3
−4
−5
−3−4−5
y
x4
5
4
3
2
1
32−1−2
−2
−3
−5
−3−4−5−6
y
x
−4
−6
−10
2
−8
−2
y
x8642−2−4−6 10
14
10
8
4
6
2
12
y
x
−4
4
−6
−8
2−2 6 8
−2
−10
y
x8642
14
12
10
8
6
4
2
−2−4−6−8
y
x−4−6−8 4
2
4
6
8
10
y
x21−1
−1
−2
−3
−2−3
3
1
y
333202CB03_AN.qxd 4/13/06 5:38 PM Page 15
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d.
Precalculus with Limits, Answers to Review Exercises 16
Review Exercises (page 271)1. 76.699 2. 361.784 3. 0.337
4. 4.181 5. 1456.529 6.
7. c 8. d 9. a 10. b
11. Shift the graph of f one unit to the right.
12. Shift the graph of f three units downward.13. Reflect f in the x-axis and shift two units to the left.14. Reflect f in the x-axis and shift eight units upward.15.
16.
17.
18.
19.
20.
21.
22.
x
2
6
8
42−2−4
y
x
6
4
2
−2
−4
−6
642−2 10
y
x
2
−2−4
6
8
2 4
y
x321−1−2−3
5
4
3
2
1
−1
y
x
−3
−6
−9
−12
−15
63−3−6 9
y
x321−1−2−3−4−5−6
1
−2
−3
−4
−5
−6
−7
−8
y
x
2
−2 2−4
4
8
4
y
�3.863
x 0 1 2
3.25 3.5 4 5 7f �x�
�1�2
0 5 6 7 8 9
3�1�3�4�4.5�4.984f �x�
x
0 2
3 �5�4.984�4.875�4f �x�
�1�2�3x
x 0 1 2 3
8 5 4.25 4.063 4.016f �x�
�1
x 0 1 2
�18.61�7.023�2.65�1�0.377f �x�
�1�2
x 0 1 2 3
4.008 4.04 4.2 5 9f �x�
�1
0 1 2
�19�7�4�3.25�3.063f �x�
�1�2x
0 1 3
0.377 1 7.0230.1420.020f �x�
�1�3x
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d.Precalculus with Limits, Answers to Review Exercises 17
(Continued)
23. 24. 25. 26.27. 2980.958 28. 1.868 29. 0.183 30. 1.32031.
32.
33.
34.
35.
36.
37. (a) 0.154 (b) 0.487 (c) 0.81138. (a) (b) $7875
(c) At the beginning. Yes39. (a) $1,069,047.14 (b) 7.9 years40. (a) 100 grams (b) 61.8 grams
(c)
41. 42.43. 44. 45. 346. 47. 48. 49.50. 51. 52. x � 6x � �5x � 5
x � 7�1�312
ln 1 � 0ln 2.2255 . . . � 0.8log25 125 �
32log4 64 � 3
Time (in years)
Mas
s of
241 P
u (i
n gr
ams) 100
80
60
40
20
20 40 60 80 100
Q
t
00
10
15,000
1 2 3 4 5
1
2
3
4
5
t
y
−6 −5 −4 −3 −2 −1 1 2
1
2
6
7
x
y
−4 −3 −1 1 2 3 4
−5
−4
−3
−2
3
x
y
−4 −3 −2 −1 1 2 3 4
2
3
4
5
6
7
x
y
x �112x �
225x � �2x � �4
x4
2
−2
−4
−6
2−4
y
0 1 2
0.35 1 1.39 1.63�0.72h�x�
�1�2x
x 0 1
0.37 1 2.72 7.39 20.09f �x�
�1�2�3
1 2 3 4
0.07 0.54 1.47 2.05 2.43s�t�
12t
n 1 2 4 12
A $6569.98 $6635.43 $6669.46 $6692.64
n 365 Continuous
A $6704.00 $6704.39
n 1 2 4 12
A $8643.88 $8799.58 $8880.43 $8935.49
n 365 Continuous
A $8962.46 $8963.38
x 0 1 2
2.72 1.65 1 0.61 0.37h�x�
�1�2
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d.
Precalculus with Limits, Answers to Review Exercises 18
(Continued)
53. Domain: 54. Domain: x-intercept: x-intercept: Vertical asymptote: Vertical asymptote:
55. Domain: 56. Domain: x-intercept: x-intercept: Vertical asymptote: Vertical asymptote:
57. Domain: 58. Domain: x-intercept: x-intercept: Vertical asymptote: Vertical asymptote:
59. 3.118 60. 61. 62. 763. 2.034 64.65. Domain: 66. Domain:
x-intercept: x-intercept: Vertical asymptote: Vertical asymptote:
67. Domain: 68. Domain: x-intercept: x-intercept: Vertical asymptote: Vertical asymptote:
69. 53.4 inches 70. 27.16 miles 71. 1.58572. 2.132 73. 74.75.
76.
77. 78.79. 80.
81. 82.
83. 84.85. 86.
87. 88. 89.
90. 91.
92. 93.
94.
95. (a)(b)
Vertical asymptote: (c) The plane is climbing at a slower rate, so the time
required increases.(d) 5.46 minutes
96. 97. 3 98.99. 100. 101. 16
102. 103. 104.
105. 106.
107. 108.
109. 110.
111. 112.ln 95ln 12
� 1.833ln 17ln 5
� 1.760
ln 20ln 6
� 1.672ln 22ln 2
� 4.459
13�ln 40 � 2� � 0.563x � 1, 3
ln 253
� 1.073ln 12 � 2.485
e�3e4 � 54.59816
ln 6ln 3 � 1.099�3s � 84.66 � 11 ln t
h � 18,000
0 20,0000
100
0 ≤ h < 18,000
ln �x � 2�5
x3�x � 2�
ln �2x � 1�x � 1�2log
1x2�x � 6�5
log8 y7 3�x � 4ln x3�x � 1�2
ln x
4�ylog6
yz2log2 5x
2 ln�y � 1� � ln 16ln�x � 3� � ln x � ln yln 3 � ln x � 2 ln y2 ln x � 2 ln y � ln z
12 log7 x � log7 41 � log3 2 �
13 log3 x
log 7 � 4 log x1 � 2 log5 xln�3� � 4 � �2.9012 ln 2 � ln 5 � 2.996
log 1 � log 3 � 2 log 2log 2
� �3.585
log 2 � 2 log 3 � 1.255�1.159�2.322
1 2 3 4 5 6
−3
−2
−1
1
2
3
x
y
x4321−1−2−3−4
4
3
2
1
−3
−4
y
x � 0x � 0�1, 0��±1, 0�
�0, �����, 0�, �0, ��
2 4 6 8
−4
−2
2
4
x
y
1−1 2 3 4 5
1
2
3
4
5
6
x
y
x � 3x � 0�4, 0��e�3, 0�
�3, ���0, ���1.530
�12�0.020
x98765
5
4
3
2
1
−2
−3
−4
−5
421−1
y
x21−1−2−3−4−6
7
6
5
4
3
2
1
y
x � 3x � �5�3.1, 0��9995, 0�
�3, ����5, ��
x8642−2
−210
8
10
6
4
2
y
x5432−1
3
2
1
−1
−2
−3
y
x � 0x � 0�10�6, 0��3, 0�
�0, ���0, ��
−1 1 2 3 4 5
−3
−2
−1
1
2
3
x
y
x4321−1−2
4
3
2
1
−1
−2
y
x � 0x � 0�1, 0��1, 0�
�0, ���0, ��
333202CB03_AN.qxd 4/13/06 5:38 PM Page 18
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d.Precalculus with Limits, Answers to Review Exercises 19
(Continued)
113.114.115. 116.
7.480; 0.392117. 118.
2.447 0.676119. 120.121. 122.123. 124.125. 126.127. No solution 128.129. 0.900 130.131. 132.
1.643 0, 0.416, 13.627
133. 134.
No solution 1.477
135. 15.2 years 136. 137. e
138. b 139. f 140. d 141. a
142. c 143. 144.
145. 2008 146. 98.6%
147. (a) 13.8629% (b) $11,486.98 148. 1243
149. (a) (b) 71
150. (a) 7.7 weeks (b) 13.3 weeks
151. watt per square centimeter
152. (a) 251,188,643 (b) 7,079,458 (c) 1,258,925,412
153. True by the inverse properties
154. False.
155. b and d are negative.a and c are positive.Answers will vary.
ln x � ln y � ln�xy� � ln�x � y�
10�3.5
40 1000
0.05
y �12e0.4605xy � 2e0.1014x
d � 221 miles
�3.990,−4
−8 16
12
−9
−5 10
1
−4
−8 16
12
−7
−4 8
1
�104�2 � �6 � 0.4495e4 � 272.991e 4 � 1 � 53.598e6�8 � 395.4293e 2 � 22.167
13e15�4 � 14.1741
4e 7.5 � 452.011
15e7.2 � 267.8861
3e8.2 � 1213.650
−10
−12 6
2
−12
−4 8
20
�1.527; �7.038−3
−12 6
9
−8
−4 11
2
ln 2 � 0.693, ln 4 � 1.386ln 5 � 1.609ln 2 � 0.693,
333202CB03_AN.qxd 4/13/06 5:38 PM Page 19
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d.
Precalculus with Limits, Answers to Chapter Test 20
Chapter Test (page 275)1. 1123.690 2. 687.291 3. 0.497 4. 22.1985.
6.
7.
8. (a) (b) 9.29.
Vertical asymptote:
10.
Vertical asymptote: 11.
Vertical asymptote: 12. 1.945 13. 0.115 14. 1.32815. 16.17.
18. 19. 20.
21. 22.
23. 24.
25. 26.27. 28. 55%29. (a)
(b) 103 centimeters; 103.43 centimeters
120110100908070605040
654321x
Hei
ght (
in c
entim
eter
s)
Age (in years)
H
y � 2745e0.1570x
800501 � 1.597e�11�4 � 0.0639
e1�2 � 1.649ln 197
4� 1.321
x �ln 44�5
� �0.757x � �2
ln x2�x � 5�
y3ln x 4
y4log3 13y
�log 7 � 2 log x� � �log y � 3 log z�ln 5 �
12 ln x � ln 6log2 3 � 4 log2 � a�
x � �6
x21−1−2−3−4−5−7
5
4
2
1
−2
−3
−4
y
x � 4
x862
4
2
−2
−4
y
x � 0
x54321−1
1
−2
−3
−4
−5
−6
−7
6 7
y
�0.89
x4321−1−2−3−4
−2
−3
−4
−5
−6
−7
y
x5431−1−2
1
−1
−2
−3
−4
−5
−6
y
x54321−1−2−3
7
1
y
x 0 1 2 3
�6�1�0.167�0.028�0.005f �x�
�1
x 0 1
0.865 0.632 0 �6.389�1.718f �x�
12�
12�1
x 1 2 4 5 6
H 58.720 75.332 86.828 103.43 110.59 117.38
14
x 1 2 4
�6.602�6.301�6.176�6�5.699f �x�
32
12
x 5 7 9 11 13
0 1.099 1.609 1.946 2.197f �x�
x 0 1
1 2.099 2.609 2.792 2.946f �x�
�1�3�5
x 0 1
10 3.162 1 0.316 0.1f �x�
12�
12�1
333202CB03_AN.qxd 4/13/06 5:38 PM Page 20
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d.Precalculus with Limits, Answers to Cumulative Test for Chapters 1–3 21
Cumulative Test for Chapters 1–3 (page 276)
1. (a) Midpoint: Distance:
2. 3.
4. 5.
6. For some values of there correspond two values of
7. (a) (b) Division by 0 is undefined. (c)
8. (a) Vertical shrink by (b) Vertical shift of two units upward(c) Horizontal shift of two units to the left
9. (a) (b) (c)
(d) Domain: all real numbers x except
10. (a) (b)(c)
(d) Domain: all real numbers x such that
11. (a) (b)Domain of all real numbers x such that Domain of all real numbers
12. (a) (b)Domain of all real numbers
13. Yes; 14. 2438.65 kilowatts
15.
16. 17.
18.
19.20.21.
22.
23.
24.
Interval:25. Intercept:
Vertical asymptotes: Horizontal asymptote:
26. y-intercept: x-intercept: Horizontal asymptote: Vertical asymptote:
27. y-intercept: x-intercepts: Slant asymptote: y � x � 4
��2, 0�, ��3, 0��0, 6�
y
x−2−3−4−5 1 2 3
2
3
4
5
(1, 0)
(0, −1)
x � �1y � 1
�1, 0��0, �1�
x6541−1−2
654321
−2−3−4−5−6
(0, 0)
y
y � 0x � ±3
�0, 0�1, 2; 1.20
−6
−3 3
4
2x3 � x2 � 2x � 10 �25
x � 2
3x � 2 �3x � 22x2 � 1
�x � 4��2x � 1��x � 1 � 3i��x � 1 � 3i�4, �12, 1 ± 3i;
�7, 0, 3; x�x��x � 3��x � 7��2, ±2i; �x � 2��x � 2i��x � 2i�
s42−2−4−6−8−10
12
10
6
4
2
−2
y
−2 −1 1 2 3 4
−3
−2
1
2
3
t
y
−8 −6 −2 2 4 6
−8
−6
−4
6
x
y
y � �34 �x � 8�2 � 5
h�1(x) �15�x � 2�
f g and g f:�x � 2��x� � 2
g f:x ≥ �6f g:
�2x2 � 62x � 12
x ≥ 1�x � 1x2 � 1
;
x2�x � 1 ��x � 1�x � 1 � x2 � 1�x � 1 � x2 � 1
x � �14
x � 34x � 1
;
4x2 � 11x � 3�3x � 45x � 2
12
s � 2
s
3
2
y.x
y � 2x � 2
−4 −2 2 4 6
−4
−2
4
6
x
y
−6 −4 2 4 6
−10
−4
−2
2
x
y
−12 −8 −4 4 8
−8
−4
8
12
16
x
y
�41�1, 32�;
333202CB03_AN.qxd 4/13/06 5:38 PM Page 21
(Continued)
Vertical asymptote:
28. or
29. All real numbers such that or
30. Reflect f in the x-axis and y-axis, and shift three units to theright.
31. Reflect f in the x-axis, and shift four units upward.
32. 1.991 33.
34. 1.717 35. 0.281
36.
37.
38.
39. or
40.
41. (a)
(b)
(c)
The model is a good fit for the data.
(d) 65.9 Yes, this is a reasonable answer.
42. 6.3 hours
7 1320
50
S � 0.274t2 � 4.08t � 50.6
7 1320
50
e6 � 2 � 401.429
3 ln 2 � 2.079ln 3 � 1.099
x �ln 12
2� 1.242
x > 0ln x2
�x � 5,
x > 4ln�x � 4� � ln�x � 4� � 4 ln x,
�0.067
−10 8
−6
f
g
6
−10 11
−7
f
g
7
x
−1−2−3−4−5−6 0 1 2
x > �1x < �5x
−2−3 3210−1
x
0 ≤ x ≤ 2x ≤ �2
y
x
(0, 6)
(−2, 0)
(−3, 0)
−6−8 2 4 6 8
2
x � �1
Precalculus with Limits, Answers to Cumulative Test for Chapters 1–3 22
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333202CB03_AN.qxd 4/13/06 5:38 PM Page 22
Precalculus with Limits, Answers to Problem Solving 23
Problem Solving (page 279)
1.
2.
increases at the fastest rate.
3. As the graph of increases at a greater rate thanthe graph of
4–6. Answers will vary.
7. (a)
(b)
(c)
8.
The pattern implies that
9.
10. 11. c
12. (a) The upper graph represents the investment compoundedannually and the lower graph represents simpleinterest.
(b)
13.
14.15. (a)
(b)(c)
(d) The exponential model is a better fit. No, because themodel is rapidly approaching infinity.
16. Answers will vary. 17.18. (a)
−4
−3 9
4
y = ln xy1
1, e2
200,0000 85
2,900,000
y2
y1
y2 � 400.88t2 � 1464.6t � 291,782y1 � 252,606�1.0310�t
B � 500�25�t�2
t �ln c1 � ln c2
� 1k2
�1k1�ln
12
Year
Inve
stm
ent (
in d
olla
rs)
5
2400
10 15 20 25 30
2000
1600
1200
800
400
y
t
y1
y2
y2 � 35t � 500y1 � 500�1.07�t
f �1�x� �
ln�x � 1x � 1�ln a
f�1�x� � ln�x � �x2 � 42 �
y
x−4 −3 −2 −1 1 2
4
3
2
1
−4
3 4
ex � 1 �x1!
�x2
2!�
x3
3!� . . .
−2
−6 6
6
y = exy4
y4 � 1 �x1!
�x2
2!�
x3
3!�
x4
4!
y = ex
y3
−2
−6 6
6
y = ex
y2
−2
−6 6
6
−2
−6 6
6
y = ex y1
xn.exx →�,
y3
00 6
24
y1
y5
y3 y2
y4
0 ≤ a ≤ 1.44y � 0.5 x and y � 1.2x
y
x−4 −3 −2 −1 1 2
7
6
5
4
3
2
−13 4
a = 0.5 a = 2
a = 1.2
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333202CB03_AN.qxd 4/13/06 5:38 PM Page 23
(Continued)
(b)
(c)
19.
The pattern implies that
20. Slope y-intercept: Slope y-intercept:
21.
17.7 cubic feet per minute
22. (a) 15 cubic feet per minute(b) 382.0 cubic feet(c) 382.0 square feet
23. (a)
(b)–(e) Answers will vary.
24. (a)
(b)–(e) Answers will vary.25. (a)
(b)–(e) Answers will vary.
26. (a)
(b)–(e) Answers will vary.
0 90
10
0 90
9
0 90
36
0 90
9
0100 1500
30
�0, ln a�� b;�0, ln a�� ln b;
ln x � �x � 1� �12�x � 1�2 �
13�x � 1�3 � . . .
−4
−3 9
4
y = ln x
y4
y4 � �x � 1� �12�x � 1�2 �
13 �x � 1�3 �
14�x � 1�4
y = ln xy3
−4
−3 9
4
y = ln x
y2
−4
−3 9
4
Precalculus with Limits, Answers to Problem Solving 24
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